According to this answer,

José Bernardo has produced an original theory of reference priors where he chooses the prior in order to maximise the information brought by the data by maximising the Kullback distance between prior and posterior. In the simplest cases with no nuisance parameters, the solution is Jeffreys' prior. In more complex problems, (a) a choice of the parameters of interest (or even a ranking of their order of interest) must be made; (b) the computation of the prior is fairly involved and requires a sequence of embedded compact sets to avoid improperness issues.

My question is this: Is there a simple example where Bernardo's reference prior, the Jeffreys prior, and the principle of transformation groups (a generalization of the principle of indifference) all exist and all yield different results?

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    $\begingroup$ As soon as you separate the parameter into groups, Jeffreys prior and reference prior differ. As for the transformation groups they only exist in some settings, a Poisson distribution being a first counterexample. $\endgroup$ – Xi'an Nov 23 '18 at 15:48
  • $\begingroup$ @Xi'an Can you clarify what you mean by "separate the parameter into groups", and by the Poisson distribution being a counterexample? $\endgroup$ – user76284 Nov 23 '18 at 18:49

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